Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Hardy's axioms

  1. Sep 19, 2003 #1
    what are they ?
    i know they are related to quantum theory.
     
  2. jcsd
  3. Sep 19, 2003 #2
  4. Sep 19, 2003 #3

    marcus

    User Avatar
    Science Advisor
    Gold Member
    2015 Award
    Dearly Missed

    I wouldnt always want to start downloading a PDF file from arxiv without first
    looking at the abstract. Some articles have hundreds of pages.
    And the title and brief summary can sometimes tell you enough. Here is the abstract for what Paden recommends reading. If you like the short summary in the abstract then click on "PDF" button right below it.

    http://arxiv.org/quant-ph/0101012
     
  5. Sep 19, 2003 #4

    marcus

    User Avatar
    Science Advisor
    Gold Member
    2015 Award
    Dearly Missed

    I'm impressed. Thanks for the link. It is Hardy's original article, only 34 pages, and gives the 5 axioms

    Here is an exerpt from Hardy's article, "Quantum Theory from Five Reasonable Axioms"
    This quote gives a taste of what it's like:

    ------------
    [[[Definition:

    The state associated with a particular preparation
    is defined to be (that thing represented by) any
    mathematical object that can be used to deter-
    mine the probability associated with the out-
    comes of any measurement that may be per-
    formed on a system prepared by the given prepa-
    ration.
    Hence, a list of all probabilities pertaining to all pos-
    sible measurements that could be made would cer-
    tainly represent the state. However, this would most
    likely over determine the state. Since most physical
    theories have some structure, a smaller set of prob-
    abilities pertaining to a set of carefully chosen mea-
    surements may be sufficient to determine the state.
    This is the case in classical probability theory and
    quantum theory.

    Central to the axioms are two inte-
    gers K and N which characterize the type of system
    being considered.

    * The number of degrees of freedom, K, is defined
    as the minimum number of probability measure-
    ments needed to determine the state, or, more
    roughly, as the number of real parameters re-
    quired to specify the state.

    * The dimension, N, is defined as the maximum
    number of states that can be reliably distinguished from one another in a single shot measurement.
    We will only consider the case where the number
    of distinguishable states is finite or countably infinite. As will be shown below, classical probability theory has K = N and quantum probability theory has K = N2 (note we do not assume that states are normalized).

    The five axioms for quantum theory (to be stated again, in context, later) are

    Axiom 1 Probabilities. Relative frequencies (mea-
    sured by taking the proportion of times a par-
    ticular outcome is observed) tend to the same
    value (which we call the probability) for any case
    where a given measurement is performed on a
    ensemble of n systems prepared by some given
    preparation in the limit as n becomes infinite.
    of N (i.e. K = K(N)) where N = 1; 2; : : : and
    where, for each given N, K takes the minimum
    value consistent with the axioms.

    Axiom 2 Simplicity. K is determined by a function
    of N (i.e. K = K(N)) where N = 1; 2; : : : and
    where, for each given N, K takes the minimum
    value consistent with the axioms.

    Axiom 3 Subspaces. A system whose state is con-
    strained to belong to an M dimensional subspace
    (i.e. have support on only M of a set of N possi-
    ble distinguishable states) behaves like a system
    of dimension M.

    Axiom 4 Composite systems. A composite system
    consisting of subsystems A and B satisfies N =
    NANB and K = KAKB

    Axiom 5 Continuity. There exists a continuous re-
    versible transformation on a system between any
    two pure states of that system.

    The first four axioms are consistent with classical
    probability theory but the fifth is not (unless the
    word "continuous" is dropped). If the last axiom is
    dropped then, because of the simplicity axiom, we
    obtain classical probability theory (with K = N) in-
    stead of quantum theory (with K = N2 ). It is very
    striking that we have here a set of axioms for quan-
    tum theory which have the property that if a single
    word is removed (namely the word "continuous" in
    Axiom 5) then we obtain classical probability theory
    instead.]]]
     
    Last edited: Sep 19, 2003
  6. Sep 19, 2003 #5

    marcus

    User Avatar
    Science Advisor
    Gold Member
    2015 Award
    Dearly Missed

    what strikes me is the
    two core definitions and the fact that in a quantum
    system the "degrees of freedom", as Hardy tells it, is equal
    to the SQUARE of the dimension
    (while in mere probability theory it is simply equal to the dimension itself) so I want to consider what he means by these two key numbers

    In the quantum case, with K = N2 , we have that

    "the minimum number of probability measurements needed to determine the state" is equal to the square of
    "the maximum
    number of states that can be reliably distinguished from one another in a single shot measurement"

    Paden Roda, any comment about where the N-squared comes from?
     
  7. Sep 19, 2003 #6
    I am nowhere near the expert in this area, but if I were to make a guess, here it goes...

    In the statement, axiom 5 states that there exists a continuous reversible transformation on a system between any two pure states of that system.

    I think the term "reversible" means that it adds a whole new set of probabilities to the system. Thus, it would give the N it's square.

    This may not make sense for 2 reasons:
    1)I'm speaking in a language that I can understand, but maybe not descriptive enough for others.
    2)Lack of knowledge on the subject.

    There are my thoughts. Make of them what you will.
    Paden Roder
     
  8. Sep 20, 2003 #7
    i agree with you.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Hardy's axioms
  1. Hardy's Paradox (Replies: 7)

  2. Hardy's paradox (Replies: 5)

  3. Hardy's Paradox (Replies: 1)

  4. QM support for Hardy (Replies: 8)

Loading...